Spheres

Help Questions

ISEE Upper Level Quantitative Reasoning › Spheres

Questions 1 - 10

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

In terms of $\pi$ , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

$\frac{9}{2} \pi \textrm{ ft}^{3}$

$9 \pi \textrm{ ft}^{3}$

$18 \pi \textrm{ ft}^{3}$

$36 \pi \textrm{ ft}^{3}$

$3 \pi \textrm{ ft}^{3}$

Explanation

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ , substitute in the volume formula, and solve for :

$V = \frac{4}{3} \pi r^{3}$

$V = \frac{4}{3} \pi\left \cdot \left( \frac{3}{2} \right ) ^{3}$

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Cannot be determined from the information provided

Explanation

A wooden ball has a surface area of $900\pi m^2$ .

What is its radius?

Begin with the formula for surface area of a sphere:

$SA_{sphere}=4\pi r^2$

Now, plug in our surface area and solve with algebra:

$900 \pi m^2=4\pi r^2$

Get rid of the pi

Divide by 4

Square root both sides to get our answer:

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

The surface area of a sphere is $4 \pi$ times the square of its radius, which here is ; the surface area of the sphere in (A) is $4 \pi t^{2}$ .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is $(2t) ^{2} = 4t^{2}$ . The cube has six faces so the total surface area is $6 \cdot 4t^{2} = 24t^{2}$ .

$\pi < 4$ , so $4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ , giving the sphere less surface area. (B) is greater.

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

$57,600 \pi\textrm{ in}^{2}$

$2,304,000 \pi \textrm{ in}^{2}$

$18,432,000\pi\textrm{ in}^{2}$

$230,400 \pi\textrm{ in}^{2}$

$307,200\pi\textrm{ in}^{2}$

Explanation

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

$A =4\pi r^{2}$

$A =4\pi \cdot 120^{2}$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

The surface area of a sphere is $4 \pi$ times the square of its radius, which here is ; the surface area of the sphere in (A) is $4 \pi t^{2}$ .

$\pi < 4$ , so $4 \pi t^{2} < 16 t^{2} < 24 t^{2}$ , giving the sphere less surface area. (B) is greater.

In terms of $\pi$ , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

$\frac{9}{2} \pi \textrm{ ft}^{3}$

$9 \pi \textrm{ ft}^{3}$

$18 \pi \textrm{ ft}^{3}$

$36 \pi \textrm{ ft}^{3}$

$3 \pi \textrm{ ft}^{3}$

Explanation

36 inches = $36 \div 12 = 3$ feet, the diameter of the tank. Half of this, or $\frac{3}{2}$ feet, is the radius. Set $r = \frac{3}{2}$ , substitute in the volume formula, and solve for :

$V = \frac{4}{3} \pi r^{3}$

$V = \frac{4}{3} \pi\left \cdot \left( \frac{3}{2} \right ) ^{3}$

$V = \frac{4}{3}\cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{1}{1}\cdot \frac{1}{1} \cdot \frac{3}{1} \cdot \frac{3}{2} \cdot \pi\left$

$V = \frac{9}{2} \pi$

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

$57,600 \pi\textrm{ in}^{2}$

$2,304,000 \pi \textrm{ in}^{2}$

$18,432,000\pi\textrm{ in}^{2}$

$230,400 \pi\textrm{ in}^{2}$

$307,200\pi\textrm{ in}^{2}$

Explanation

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

$A =4\pi r^{2}$

$A =4\pi \cdot 120^{2}$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

There is a perfectly spherical weather balloon with a surface area of $900 \pi ft^2$ , what is its diameter?